Two-dimensional Modal Logics of Relational Algebras and Temporal Logics of Intervals

The central idea of the paper is to look at both relation algebras (cf. [JT], [J]) and temporal logic of intervals ([vB1], [V]) from a viewpoint of twodimensional modal logic. Here we mean by a two-dimensional modal logic a system of modal logic in which (at least some of) the intended Kripke semantics is such that the set of possible worlds is (a subset of) a Cartesian product A×A for some set A: i.e. possible worlds are pairs of more basic objects. The report originated with the simple observation that if one views an interval as the ordered pair consisting of its beginingand endpoint, a set of intervals can be seen as a binary relation (on the set of timepoints). We define a logic CC of which the associated modal algebras have the type of relation algebras, and a modal logic of intervals CDT which has a CHOP -operator C.